$(A\cdot x)\cdot (B\cdot x)=(A\cdot x)\cdot (B\cdot x)= (B\cdot x)^t * (A \cdot x)=(x^t \cdot B^t) * (A\cdot x)=x^t \cdot (B^t * A)\cdot x$ Correct?

Question

Let $A,B$ be $m\times n$-matrices and $x\in \mathbb{R}^n$ then $(A\cdot x)\cdot (B\cdot x)=(A\cdot x)\cdot (B\cdot x)= (B\cdot x)^t * (A \cdot x)=(x^t \cdot B^t) * (A\cdot x)=x^t \cdot (B^t * A)\cdot x$

Let $*$ denote the matrix product and $\cdot$ the scalar product. Let $x,y\in \mathbb{R}^n$ then $x\cdot y=y^t * x$.

Are my equivalence formulations correct?